Given an endofunctor \(\mathcal {F}:\mathcal {C}\rightarrow \mathcal {C}\)  on a category \(\mathcal {C}\) , an \(\mathcal {F}\) -coalgebra is a
\(\mathcal {C}\) -object \(X\)  together with a \(\mathcal {C}\) -arrow \(f:X \rightarrow \mathcal {F}X\) . In many categories
the pair \((X,f)\)  represents a transition system such as a deterministic,
nondeterministic or probabilistic automaton [1]}. A
morphism of \(\mathcal {F}\) -coalgebras, or an \(\mathcal {F}\) -homomorphism, between
coalgebras \((X,f)\)  and \((Y,g)\)  is a \(\mathcal {C}\) -arrow \(h: X \rightarrow Y\)  such that the
following diagram commutes.
\(\begin{tikzcd}X [r, "h"] [d, "f"^{\prime }] & Y [d, "g"] \\\mathcal {F}X [r, "Fh"] & \mathcal {F}Y\end{tikzcd}\) 
